Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2
Lesson 4: Definition of Reflection and Basic Properties
Student Outcomes

Students know the definition of reflection and perform reflections across a line using a transparency.

Students show that reflections share some of the same fundamental properties with translations (e.g., lines
map to lines, angle- and distance-preserving motion). Students know that reflections map parallel lines to
parallel lines.

Students know that for the reflection across a line 𝐿 and for every point 𝑃 not on 𝐿, 𝐿 is the bisector of the
segment joining 𝑃 to its reflected image 𝑃′.
Classwork
Example 1 (5 minutes)
The reflection across a line 𝐿 is defined by using the following example.
MP.6

Let 𝐿 be a vertical line, and let 𝑃 and 𝐴 be two points not on 𝐿, as shown below. Also, let 𝑄 be a point on 𝐿.
(The black rectangle indicates the border of the paper.)

The following is a description of how the reflection moves the points 𝑃, 𝑄, and 𝐴 by making use of the
transparency.

Trace the line 𝐿 and three points onto the transparency exactly, using red. (Be sure to use a transparency that
is the same size as the paper.)

Keeping the paper fixed, flip the transparency across the vertical line (interchanging left and right) while
keeping the vertical line and point 𝑄 on top of their black images.

The position of the red figures on the transparency now represents the
reflection of the original figure. 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝑃) is the point represented by the
red dot to the left of 𝐿, 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐴) is the red dot to the right of 𝐿, and point
𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝑄) is point 𝑄 itself.

Note that point 𝑄 is unchanged by the reflection.
Lesson 4:
Definition of Reflection and Basic Properties
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Scaffolding:
There are manipulatives, such
as MIRA and Georeflector,
which facilitate the learning of
reflections by producing a
reflected image.
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2

The red rectangle in the picture below represents the border of the transparency.

In the picture above, the reflected image of the points is noted similar to how we represented translated
images in Lesson 2. That is, the reflected point 𝑃 is 𝑃′. More importantly, note that line 𝐿 and point 𝑄 have
reflected images in exactly the same location as the original; hence, 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿) = 𝐿 and 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝑄) =
𝑄, respectively.

The figure and its reflected image are shown together, below.

Pictorially, reflection moves all of the points in the plane by reflecting them across 𝐿 as if 𝐿 were a mirror. The
line 𝐿 is called the line of reflection. A reflection across line 𝐿 may also be noted as 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝐿 .
Video Presentation (2 minutes)
1
The following animation of a reflection is helpful to beginners.
http://www.harpercollege.edu/~skoswatt/RigidMotions/reflection.html
1
Animation developed by Sunil Koswatta.
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2
Exercises 1–2 (3 minutes)
Students complete Exercises 1 and 2 independently.
Exercises
1.
Reflect △ 𝑨𝑩𝑪 and Figure 𝑫 across line 𝑳. Label the reflected images.
2.
Which figure(s) were not moved to a new location on the plane under this transformation?
Point 𝑩 and line 𝑳 were not moved to a new location on the plane under this reflection.
Lesson 4:
Definition of Reflection and Basic Properties
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 4
8•2
Example 2 (3 minutes)
Now the lesson looks at some features of reflected geometric figures in the plane.

If we reflect across a vertical line 𝑙, then the reflected image of right-pointing figures, such as 𝑇 below, are leftpointing. Similarly, the reflected image of a right-leaning figure, such as 𝑆 below, becomes left-leaning.

Observe that up and down do not change in the reflection across a vertical line. Also observe that the
horizontal figure 𝑇 remains horizontal. This is similar to what a real mirror does.
Example 3 (2 minutes)
A line of reflection can be any line in the plane. This example looks at a horizontal line of reflection.

Let 𝑙 be the horizontal line of reflection, 𝑃 be a point off of line 𝑙, and 𝑇 be the figure above the line of
reflection.

Just as before, if we trace everything in red on the transparency and reflect across the horizontal line of
reflection, we see the reflected images in red, as shown below.
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2
Exercises 3–5 (5 minutes)
Students complete Exercises 3–5 independently.
3.
Reflect the images across line 𝑳. Label the reflected images.
4.
Answer the questions about the image above.
a.
Use a protractor to measure the reflected ∠𝑨𝑩𝑪. What do you notice?
The measure of the reflected image of ∠𝑨𝑩𝑪 is 𝟔𝟔°.
b.
Use a ruler to measure the length of 𝑰𝑱 and the length of the image of 𝑰𝑱 after the reflection. What do you
notice?
The length of the reflected segment is the same as the original segment, 𝟓 units.
Note: This is not something students are expected to know, but it is a preview for what is to come later in this
lesson.
5.
Reflect Figure 𝑹 and △ 𝑬𝑭𝑮 across line 𝑳. Label the reflected images.
Lesson 4:
Definition of Reflection and Basic Properties
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 4
8•2
Discussion (3 minutes)
As with translation, a reflection has the same properties as (Translation 1)–(Translation 3) of Lesson 2. Precisely, lines,
segments, and angles are moved by a reflection by moving their exact replicas (on the transparency) to another part of
the plane. Therefore, distances and degrees are preserved.
(Reflection 1) A reflection maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an
angle.
(Reflection 2) A reflection preserves lengths of segments.
(Reflection 3) A reflection preserves degrees of angles.
These basic properties of reflections are taken for granted in all subsequent discussions of geometry.
Basic Properties of Reflections:
(Reflection 1) A reflection maps a line to a line, a ray to a ray, a segment to a segment, and an angle to an
angle.
(Reflection 2) A reflection preserves lengths of segments.
(Reflection 3) A reflection preserves measures of angles.
If the reflection is across a line 𝑳 and 𝑷 is a point not on 𝑳, then 𝑳 bisects and is perpendicular to the segment 𝑷𝑷′,
joining 𝑷 to its reflected image 𝑷′. That is, the lengths of 𝑶𝑷 and 𝑶𝑷′ are equal.
Example 4 (7 minutes)
A simple consequence of (Reflection 2) is that it gives a more precise description of the position of the reflected image of
a point.

Let there be a reflection across line 𝐿, let 𝑃 be a point not on line 𝐿, and let 𝑃′ represent 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝑃). Let
the line 𝑃𝑃′ intersect 𝐿 at 𝑂, and let 𝐴 be a point on 𝐿 distinct from 𝑂, as shown.
Lesson 4:
Definition of Reflection and Basic Properties
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
Lesson 4
8•2
What can we say about segments 𝑃𝑂 and 𝑂𝑃′ ?

Because 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝑃𝑂) = 𝑃′𝑂, (Reflection 2) guarantees that segments 𝑃𝑂 and 𝑃′𝑂 have the same
length.

In other words, 𝑂 is the midpoint (i.e., the point equidistant from both endpoints) of 𝑃𝑃′.

In general, the line passing through the midpoint of a segment is said to bisect the segment.

What happens to point 𝐴 under the reflection?

Because the line of reflection maps to itself, point A remains unmoved, that is, 𝐴 = 𝐴′ .

As with translations, reflections map parallel lines to parallel lines (i.e., if 𝐿1 ∥ 𝐿2 , and there is a reflection
across a line, then 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿1 ) ∥ 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿2 )).

Let there be a reflection across line 𝑚. Given 𝐿1 ∥ 𝐿2 , then 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿1 ) ∥ 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿2 ). The reason is
that any point 𝐴 on line 𝐿1 will be reflected across 𝑚 to a point 𝐴′ on 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿1 ). Similarly, any point 𝐵
on line 𝐿2 will be reflected across 𝑚 to a point 𝐵′ on 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿2 ). Since 𝐿1 ∥ 𝐿2 , no point 𝐴 on line 𝐿1 will
ever be on 𝐿2 , and no point 𝐵 on 𝐿2 will ever be on 𝐿1 . The same can be said for the reflections of those
points. Then, since 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿1 ) shares no points with 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿2 ), 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿1 ) ∥
𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐿2 ).
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2
Exercises 6–9 (7 minutes)
Students complete Exercises 6–9 independently.
Use the picture below for Exercises 6–9.
6.
Use the picture to label the unnamed points.
Points are labeled in red above.
7.
What is the measure of ∠𝑱𝑲𝑰? ∠𝑲𝑰𝑱? ∠𝑨𝑩𝑪? How do you know?
𝒎∠𝑱𝑲𝑰 = 𝟑𝟏°, 𝒎∠𝑲𝑰𝑱 = 𝟐𝟖°, and 𝒎∠𝑨𝑩𝑪 = 𝟏𝟓𝟎°. Reflections preserve angle measures.
8.
What is the length of segment 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑭𝑯)? 𝑰𝑱? How do you know?
|𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑭𝑯)| = 𝟒 units, and 𝑰𝑱 = 𝟕 units. Reflections preserve lengths of segments.
9.
What is the location of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑫)? Explain.
Point 𝑫 and its image are in the same location on the plane. Point 𝑫 was not moved to another part of the plane
because it is on the line of reflection. The image of any point on the line of reflection will remain in the same
location as the original point.
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
8•2
Closing (4 minutes)
Summarize, or have students summarize, the lesson.

We know that a reflection across a line is a basic rigid motion.

Reflections have the same basic properties as translations; reflections map lines to lines, rays to rays, segments
to segments and angles to angles.

Reflections have the same basic properties as translations because they, too, are distance- and anglepreserving.

The line of reflection 𝐿 is the bisector of the segment that joins a point not on 𝐿 to its image.
Lesson Summary

A reflection is another type of basic rigid motion.

A reflection across a line maps one half-plane to the other half-plane; that is, it maps points from one
side of the line to the other side of the line. The reflection maps each point on the line to itself. The
line being reflected across is called the line of reflection.

When a point, 𝑷 is joined with its reflection, 𝑷′ to form segment 𝑷𝑷′, the line of reflection bisects and
is perpendicular to the segment 𝑷𝑷′ .
Terminology
REFLECTION (description): Given a line 𝑳 in the plane, a reflection across 𝑳 is the transformation of the plane that
maps each point on the line 𝑳 to itself, and maps each remaining point 𝑷 of the plane to its image 𝑷′ such that 𝑳 is
the perpendicular bisector of the segment 𝑷𝑷′ .
Exit Ticket (4 minutes)
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
8•2
Date
Lesson 4: Definition of Reflection and Basic Properties
Exit Ticket
1.
Let there be a reflection across line 𝐿𝐴𝐵 . Reflect △ 𝐶𝐷𝐸 across line 𝐿𝐴𝐵 . Label the reflected image.
2.
Use the diagram above to state the measure of 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(∠𝐶𝐷𝐸). Explain.
3.
Use the diagram above to state the length of segment 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛(𝐶𝐸). Explain.
4.
Connect point 𝐶 to its image in the diagram above. What is the relationship between line 𝐿𝐴𝐵 and the segment that
connects point 𝐶 to its image?
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2
Exit Ticket Sample Solutions
1.
Let there be a reflection across line 𝑳𝑨𝑩 . Reflect △ 𝑪𝑫𝑬 across line 𝑳𝑨𝑩 . Label the reflected image.
2.
Use the diagram above to state the measure of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(∠𝑪𝑫𝑬). Explain.
The measure of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(∠𝑪𝑫𝑬) is 𝟗𝟎° because reflections preserve degrees of measures of angles.
3.
Use the diagram above to state the length of segment 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑪𝑬). Explain.
The length of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑪𝑬) is 𝟏𝟎 𝐜𝐦 because reflections preserve segment lengths.
4.
Connect point 𝑪 to its image in the diagram above. What is the relationship between line 𝑳𝑨𝑩 and the segment that
connects point 𝑪 to its image?
The line of reflection bisects the segment that connects 𝑪 to its image.
Lesson 4:
Definition of Reflection and Basic Properties
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Lesson 4
NYS COMMON CORE MATHEMATICS CURRICULUM
8•2
Problem Set Sample Solutions
1.
In the picture below, ∠𝑫𝑬𝑭 = 𝟓𝟔°, ∠𝑨𝑪𝑩 = 𝟏𝟏𝟒°, 𝑨𝑩 = 𝟏𝟐. 𝟔 units, 𝑱𝑲 = 𝟓. 𝟑𝟐 units, point 𝑬 is on line 𝑳, and
point 𝑰 is off of line 𝑳. Let there be a reflection across line 𝑳. Reflect and label each of the figures, and answer the
questions that follow.
2.
What is the measure of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(∠𝑫𝑬𝑭)? Explain.
The measure of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(∠𝑫𝑬𝑭) is 𝟓𝟔°. Reflections preserve degrees of angles.
3.
What is the length of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑱𝑲)? Explain.
The length of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑱𝑲) is 𝟓. 𝟑𝟐 units. Reflections preserve lengths of segments.
4.
What is the measure of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(∠𝑨𝑪𝑩)?
The measure of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(∠𝑨𝑪𝑩) is 𝟏𝟏𝟒°.
5.
What is the length of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑨𝑩)?
The length of 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏(𝑨𝑩) is 𝟏𝟐. 𝟔 units.
Lesson 4:
Definition of Reflection and Basic Properties
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6.
Lesson 4
8•2
Two figures in the picture were not moved under the reflection. Name the two figures, and explain why they were
not moved.
Point 𝑬 and line 𝑳 were not moved. All of the points that make up the line of reflection remain in the same location
when reflected. Since point 𝑬 is on the line of reflection, it is not moved.
7.
Connect points 𝑰 and 𝑰′. Name the point of intersection of the segment with the line of reflection point 𝑸. What do
you know about the lengths of segments 𝑰𝑸 and 𝑸𝑰′?
Segments 𝑰𝑸 and 𝑸𝑰′ are equal in length. The segment 𝑰𝑰′ connects point 𝑰 to its image, 𝑰′. The line of reflection
will go through the midpoint, or bisect, the segment created when you connect a point to its image.
Lesson 4:
Definition of Reflection and Basic Properties
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